Number Sense Research Articles

Unveiling the numerical insights: A study on number sense skills of first graders and their teachers

This study explores the number sense skills of classroom teachers and first-grade primary school students in Uşak, Turkey, examining the impact of various demographic factors. A quantitative research design was employed, utilizing a survey model to gather data from 102 teachers and 341 students. The Number Sense Test (Teacher Form), developed by Kayhan Altay (2010), and the Number Sense Screener (Student Form), adapted by Uyanık Aktulun (2018) based on Jordan et al. (2010), were used as measurement instruments. The results revealed significant differences in number sense skills among teachers based on gender, professional experience, and training, and among students based on gender and parental education levels. Male teachers exhibited higher number sense skills compared to female teachers, and those with more professional experience and specific number sense training performed better. Among students, males outperformed females in certain number sense subcomponents, and higher parental education levels were associated with better student performance. These findings highlight the importance of targeted professional development for teachers and the crucial role of parental education levels in children's numerical development. The study underscores the need for early intervention programs and parental involvement to foster strong number sense skills, which are foundational for later mathematical achievement.

Number adaptation: Reply

Adaptation is a ubiquitous property of perceptual systems, increasing sensitivity to change and allowing them to operate over a large dynamic range. The number sense, like most other perceptual systems, is adaptable. Yousif et al. (2024) challenge the existence of number adaptation, offering an alternate explanation that they term the “old news hypothesis”. Here we consider the major evidence advanced for their theory and show that, while their predicted effects may reach statistical significance, they are far too small to begin to explain the robust phenomenon of adaptation. We also highlight a series of studies using fMRI, EEG, pupillometry and psychophysical techniques that support the existence of adaption, and are inconsistent with “old news”. We conclude that number adaptation, while not fully understood, does indeed exist, and remains an invaluable concept for understanding the number sense.

Investigating the development of number sense in a 6th grade class in Taiwan

Investigating the development of number sense in a 6th grade class in Taiwan

An introduction to early grade mental mathematics and number sense

An introduction to early grade mental mathematics and number sense

Timing matters: investigating the interplay of presentation duration and congruency in approximate number processing.

The approximate number system (ANS) is an innate number sense ability; it plays a pivotal role in the development of symbolic number ability. Despite studies using a wide range of presentation durations of stimuli to investigate the approximate number processing, limited literature has systematically explored its impact on approximate number processing. Further, if increasing presentation duration leads to improved accuracy, it remains unclear whether this improvement will be driven by better performance in both congruent and incongruent conditions or only in one condition. Addressing these gaps, the present study investigated the impact of presentation duration on the approximate number processing among 45 college-going adult participants from India. The dot number comparison task was used to measure ANS, with stimuli presentation duration manipulated at the following durations: 100, 200, 500, 1000, and 3000 milliseconds (or until response). Based on the congruency of non-numerical features of dots with numerosity of dots, trials were categorized into congruent and incongruent trials. Repeated measure ANOVA revealed a positive relationship between the presentation duration and dot number comparison accuracy. However, a saturation point in performance was observed at a presentation duration of 500 milliseconds, beyond which increased duration did not lead to enhanced accuracy. Further analysis based on congruency revealed that performance improvement was confined only to congruent conditions. Drawing on samples from India, the study offers valuable insights into the ANS.

Mature number sense predicts middle school students’ mathematics achievement.

Mature number sense predicts middle school students’ mathematics achievement.

The Mechanisms and Neural Signature of Time-averaged Numerosity Perception.

The animal brain is endowed with an innate sense of number allowing to intuitively perceive the approximate quantity of items in a scene, or "numerosity." This ability is not limited to items distributed in space, but also to events unfolding in time and to the average numerosity of dynamic scenes. How the brain computes and represents the average numerosity over time, however, remains unclear. Here, we investigate the mechanisms and EEG signature of the perception of average numerosity over time. To do so, we used stimuli composed of a variable number (3-12) of briefly presented dot arrays (50 msec each) and asked participants to judge the average numerosity of the sequence. We first show that the weight of different portions of the stimuli in determining the judgment depends on how many arrays are included in the sequence itself: the longer the sequence, the lower the weight of the latest arrays. Second, we show systematic adaptation effects across stimuli in consecutive trials. Importantly, the EEG results highlight two processing stages whereby the amplitude of occipital ERPs reflects the adaptation effect (∼300 msec after stimulus onset) and the accuracy and precision of average numerosity judgments (∼450-700 msec). These two stages are consistent with processes involved with the representation of perceived average numerosity and with perceptual decision-making, respectively. Overall, our findings provide new evidence showing how the visual system computes the average numerosity of dynamic visual stimuli, and support the existence of a dedicated, relatively low-level perceptual mechanism mediating this process.

The work of a prospective high school teacher in pre-professional training in highlighting mathematical knowledge

This article aims to identify the knowledge that enables a prospective secondary school teacher to mobilize number sense components during a pre-professional practicum. Using a qualitative case study approach, four classroom episodes focused on fractions, percentages, and algebraic language are analyzed. The study involves one prospective teacher and 23 students from a multi-grade school in central-southern Chile. The analysis integrates teacher’s specialized knowledge and mathematical working spaces to explore how these frameworks inform the teacher’s knowledge. The findings reveal that the teacher employs personal theories, real-life examples, and graphical aids to enhance students’ understanding, though there is a reliance on rules and algorithms. While specialized knowledge aids in teaching, there is less emphasis on fostering students’ conceptual understanding and judgement. The study recommends improving teacher training programs to better equip prospective teachers in grasping number sense components, highlighting the need for more focused training on its development.

Investigating fraction computation problem-solving among pre-service primary school teachers

The ability to solve problems involving fractions is a fundamental aspect of mathematics education. This study explores how Pre-Service Primary School Teachers approach problem-solving in fractional computations. A workbook is designed to support pre-service primary school teachers ' computational thinking in fraction-context challenges. The study employs a qualitative descriptive method encompassing interviews, study documentation, and observation to assess fraction computation problem-solving abilities. Twenty-seven participants were involved in this study. They were first-year students enrolled in the Primary School Teacher Education Department at one of the private universities in Jakarta, Indonesia. The findings reveal a notable outcome in pre-service primary school teachers ' understanding of fraction computation problem-solving, marked by recognizable strategies in their problem-solving approach. This research suggests that designing a series of workbooks containing various strategies in computational fractions and building a strong fractional number sense can help pre-service teachers reduce misconceptions and better understand fraction operations. These findings offer guidance for mathematics teacher education on how to effectively teach and embed the concept of fraction calculations to their future students so that they can only teach procedurally if they understand the meaning of fraction operations.

The divisive normalization model of visual number sense: model predictions and experimental confirmation.

Our intuitive sense of number allows rapid estimation for the number of objects (numerosity) in a scene. How does the continuous nature of neural information processing create a discrete representation of number? A neurocomputational model with divisive normalization explains this process and existing data; however, a successful model should not only explain existing data but also generate novel predictions. Here, we experimentally test novel predictions of this model to evaluate its merit for explaining mechanisms of numerosity perception. We did so by consideration of the coherence illusion: the underestimation of number for arrays containing heterogeneous compared to homogeneous items. First, we established the existence of the coherence illusion for homogeneity manipulations of both area and orientation of items in an array. Second, despite the behavioral similarity, the divisive normalization model predicted that these two illusions should reflect activity in different stages of visual processing. Finally, visual evoked potentials from an electroencephalography experiment confirmed these predictions, showing that area and orientation coherence modulate brain responses at distinct latencies and topographies. These results demonstrate the utility of the divisive normalization model for explaining numerosity perception, according to which numerosity perception is a byproduct of canonical neurocomputations that exist throughout the visual pathway.

Practical Pedagogical Approaches: Integrating Play-based and Experiential Learning at Pre-Primary Education as per NEP 2020 and NCF-FS 2022

Early childhood education in India is undergoing significant changes, guided by the National Education Policy (NEP) 2020 and the National Curriculum Framework for the Foundational Stage (NCF-FS) 2022. This paper explores practical pedagogical approaches for pre-primary and early childhood care and education (ECCE) in India, aligning with these policy recommendations. It examines key strategies for implementing play-based learning, developing language and literacy skills, building foundational numeracy competencies, and incorporating inquiry-based and activity-based learning approaches. The study emphasizes the importance of creating stimulating learning environments featuring well-equipped indoor and outdoor play areas to facilitate meaningful play experiences. It details methods for fostering oral language development, phonological awareness, and early literacy skills through various activities and a print-rich classroom setting. Strategies for developing number sense, basic operations, spatial awareness, and other core numeracy concepts are outlined. The paper also explores approaches for integrating inquiry-based learning to nurture scientific thinking and problem-solving skills. Additionally, the paper discusses the role of diverse activities in promoting holistic development across physical, cognitive, social-emotional, language, and aesthetic domains. It highlights integrating art and experimentation in ECCE to enhance creativity and self-expression. The role of educators is analyzed, focusing on their contribution to creating stimulating learning environments and facilitating meaningful experiences for children. Challenges in implementing these approaches include infrastructure and resource constraints, teacher training needs, parental engagement, and cultural relevance. However, the evolving policy landscape also presents opportunities for transformation through increased investment, flexibility in curriculum design, and an emphasis on developmentally appropriate practices. The paper concludes with recommendations for future directions, such as strengthening ECCE infrastructure, enhancing teacher professional development, fostering community engagement, conducting further research, and developing culturally appropriate resources. By synthesizing evidence-based practices aligned with policy directives, the paper provides a comprehensive overview of effective pedagogy for fostering young children’s holistic growth and development in early childhood education in India.

EXAMINING NUMBER SENSE SKILLS OF KINDERGARTEN CHILDREN

The aim of this study is to determine the number sense skills of preschool (Kindergarten) children. In the study, survey design, which is one of the quantitative research methods, was used. The study population of the research consists of a total of 114 children attending all of kindergartens (5 kindergartens) in the city center of Tunceli in the 2020-2021 academic year. The data collection tool used in the study is the "Preschool Number Sense Test" developed by the researchers. In the analysis of the data, the items in the number sense test; The correct answer obtained by using number sense skills, the correct answer obtained by using rule and operation-based methods, the wrong answer obtained by using number sense skills, the wrong answer obtained by using rule-operation-based solutions, and the blank answer were coded. As a result of the analysis made, it was determined that preschool students used rule- and operation-based solutions more in solving questions, and it was seen that they resorted to number-sense-based solutions less frequently.

The development of pre-service teachers’ competence to teach mental calculation strategies

Background: There is a concern in South Africa that pre-service teachers do not have the required knowledge to teach mathematics in primary school and to develop learners’ number sense. In this study, pre-service teachers taught the mental strategy of bridging through ten through a structured teaching sequence from the Mental Starters Assessment Project (MSAP) materials as a work-integrated learning opportunity.Aim: We ask the question: How do the MSAP materials support pre-service teachers in competently teaching mental mathematics?Setting: Thirty-eight Bachelor of Education (Foundation Phase) third-year preservice teachers from an Eastern Cape university participated in this study.Methods: Participants taught the strategy during their Teaching Practice, quantitatively analysed the results of their classes and reflected on the experience in a questionnaire and focus group interviews.Results: Results indicate that the teachers were relatively successful in their teaching of the strategy; however, all indicated that they taught the sequence for a more extended period than recommended. Qualitative responses provide evidence of the teachers’ development in their knowledge of learners and their characteristics, general pedagogical knowledge, pedagogical content knowledge and knowledge of educational contexts.Conclusion: This study offers evidence of the professional learning of pre-service teachers that resulted from taking an integrated approach to facilitating a mathematics teaching methodology course through requiring a work-integrated learning component.Contribution: We argue that such an approach is necessary for pre-service teachers to be adequately prepared for the challenges of teaching mathematics in the South African classroom.

Development and validation of a two‐tier number sense test for sixth‐grade students in fractions and decimals

AbstractThe aim of this study was to develop an instrument for assessing sixth‐grade students’ number sense skills in fractions and decimals. This study was conducted on 452 sixth graders (10–11 years old) from the western region of Turkey. The construct validity of the number sense test (NST) was examined via exploratory factor analysis (EFA) and confirmatory factor analysis (CFA). First, the draft NST, which consists of 36 two‐tier items, was administered to 208 students for the EFA. Repeated EFA results indicated that the NST consisted of five number sense components with 25 items (understanding the meaning and magnitude of numbers, recognizing the effects of operations on numbers, using multiple representations of numbers, using reference points in numbers, and judging the reasonableness of a computational result). The NST, with factor loadings ranging from 0.60 to 0.87, accounted for 72.39% of the total variance. The composite reliability and Cronbach's alpha coefficients ranged from 0.88 to 0.92. The CFA was administered to an additional 244 students in the school. The results of the first‐ and second‐order CFA revealed that the model fit of the NST was good (Chi‐square/df = 2.04; RMSEA = 0.066; NNFI = 0.97; IFI = 0.98; CFI= 0.97; and S‐RMR = 0.065). The results of the present study indicate that the NST is a valid and reliable instrument for assessing sixth‐grade students' number sense skills in fractions and decimals.

Neural Basis of Number Sense in Larval Zebrafish.

Number sense, the ability to discriminate the quantity of objects, is crucial for survival. To understand how neurons work together and develop to mediate number sense, we used two-photon fluorescence light sheet microscopy to capture the activity of individual neurons throughout the brain of larval Danio rerio, while displaying a visual number stimulus to the animal. We identified number-selective neurons as early as 3 days post-fertilization and found a proportional increase of neurons tuned to larger quantities after 3 days. We used machine learning to predict the stimulus from the neuronal activity and observed that the prediction accuracy improves with age. We further tested ethanol's effect on number sense and found a decrease in number-selective neurons in the forebrain, suggesting cognitive impairment. These findings are a significant step towards understanding neural circuits devoted to discrete magnitudes and our methodology to track single-neuron activity across the whole brain is broadly applicable to other fields in neuroscience.

Typical Mathematical Education Cannot Predict Non‐Verbal Number Sense

ABSTRACTFormal education would promote symbolic number processing ability, but the relationship between non‐symbolic number sense and mathematical education remains controversial. The current investigation hypothesized that non‐symbolic number sense is independent from the limited human experience of mathematical education, while symbolic number processing could be influenced by related closely to mathematical education. Experiment 1 compared both non‐symbolic number sense and arithmetic computation of undergraduates majoring in STEM fields and humanities. Experiment 2 compared homologous undergraduates from China and Kyrgyzstan in non‐symbolic number sense and mathematical cognitive processing. Both of two experiments found that mathematical education was significantly associated with symbolic mathematics, but not with non‐symbolic number sense. The results suggest that while mathematical education might improve symbolic mathematics, it does not alter non‐symbolic number sense.

Number Talks as Access to Teaching Practices: Pre-Service teachers’ Attention to Mathematical Teaching Practices While Reflecting on Number Talks Enacted in a Field Placement

ABSTRACT Number Talks are increasingly used in classrooms to support student’s number sense. While Matney et al. (2020) documented little empirical evidence about the effectiveness of Number Talks in improving student outcomes, we considered the potential benefits of Number Talks for teachers and their practices. We ask to which of NCTM’s Mathematical Teaching Practices do pre-service teachers attend while reflecting on Number Talks implemented in a field placement and what was the nature of their attention? Using a constructivist lens, we analyzed the reflections of ten pre-service elementary teachers on three Number Talks in a field placement. As they implemented Number Talks, the participants in this study constructed their own understanding of both the process and purpose of a Number Talk. By doing so, we found that not only was it possible for pre-service teachers to attend to each of the Mathematical Teaching Practices, but three practices were regularly attended to by participants: posing purposeful questions, facilitating mathematical discourse, and supporting productive struggle. Thus, Number Talks may serve pre-service teachers as important vehicles for developing these practices.

Pre-service teachers’ use of the jump strategy on the empty number line for mental computation

Background: South African teachers’ mathematical knowledge for teaching and number sense are markedly low, evidenced in national and international research. There is a call for supporting teachers in developing their knowledge and confidence in teaching mathematics.Aim: The aim of this article is to share how pre-service teachers (PSTs) used the empty number line (ENL) to demonstrate the jump strategy (JS) for efficient mental computation, related to the national Mental Starters Assessment Project (MSAP).Setting: This study was conducted at a tertiary institution in the Eastern Cape province of South Africa, with 40 first-year intermediate phase PSTs, during their mathematics didactics course.Methods: The data comprised 40 micro-teaching videos, analysed thematically, following an analytical model for video-recorded data. Videos were categorised according to the way they faithfully followed or departed from the MSAP teacher resources.Results: Findings suggest that more than 70% of the PSTs faithfully followed the teacher resources to teach the JS on the ENL, with regards to use of at least three of the four key elements, namely: ENL, jumps, gesturing, and key phrases and terminology.Conclusion: We suggest PSTs receive explicit instruction on the above-mentioned key elements for teaching mental calculation strategies with the ENL and opportunities to practise teaching these to their peers.Contribution: Most PSTs successfully used the ENL to demonstrate the JS for efficient mental computation, enabled by the opportunity to engage in developing (and reflecting on their) micro-teaching videos.

Number Sense: Flexibility in Thinking about Numbers

The aim of this research is to understand the flexibility of students in solving problems related to number sense. Flexibility can be observed through the strategies used by students, whether they employ standard algorithmic procedures or utilize intuitive thinking strategies. This study is a qualitative research with a case study method. The research sample consists of 35 eighth-grade students who have covered the material of integers and have been given a written test. Test results are divided into two categories: students who solve problems using number sense strategies and students who do not use number sense strategies. In-depth interviews were conducted with two representative students who took the written test. The interview results indicate that students who solve problems with number sense strategies understand the concept of number density, while students who do not use number sense strategies do not yet have an understanding of the meaning of numbers. Overall, students tend to use procedural methods to solve the given problems. The results of this research are expected to be used as a basis for designing learning activities that accommodate number sense abilities.

“Awur Eser” to Explore Number Sense: Hypothetical Learning Trajectory for Integer Material Facilitating Learning Speed

This article aims to explore the concept of "Awur Eser" as a potential learning method to enhance students' number sense, especially in the context of learning integer concepts, and to develop more effective and inclusive strategies for mathematics education. We conducted a literature review to identify students’ barriers to understanding integers based on their number sense and learning speed. Then, we conducted a second literature review to design a Hypothetical Learning Trajectory (HLT) for integer learning. Subsequently, we held a Focus Group Discussion (FGD) with mathematics education experts and teachers to discuss and validate the selection of HLT for integer learning. Based on the FGD results, we compiled the final design of HLT for integer learning. This design involves adjustments based on input from experts and teachers, ensuring alignment with students' characteristics and needs. The implementation of HLT hopefully can improve students' conceptual understanding and accelerate the learning process.